For shortness, let
\(G:=\mathrm {Int}(A).\) 
We define
\(\widetilde{u}_h := (u_{m_h} + a_h)\chi _{A_{m_h}}^{} + \eta \chi _{\Omega \setminus A_{m_h}}\) 
and
\(\widetilde{u}:= u\chi _G^{} + \eta \chi _{\Omega \setminus G}^{}\) 
for \(\eta \in (0,1)^2\)  such that \(\Omega \cap \partial ^*A_{m_h}\subset J_{\widetilde{u}_h}\)  and \(\Omega \cap \partial ^*G\subset J_{\widetilde{u}}\)  up to an \(\mathcal {H}^1\) -negligible set. Such \(\eta \)  exists by Lemma REF  in view of the estimate
\(\mathcal {H}^1(\partial A_{m_h}) \le \frac{\mathcal {S}(A_{m_h},u_{m_h})}{c_1} + \frac{2c_2\mathcal {H}^1(\partial \Omega )}{c_1}\le \frac{\mathcal {F}(A_1,u_1)}{c_1} + \frac{2c_2\mathcal {H}^1(\partial \Omega )}{c_1},\) 
which holds for every \(h\ge 1.\) 
Notice that \(\widetilde{u}_h\in GSBD^2({\rm Int}{(\Omega \cup S\cup \Sigma )})\cap H_\mathrm {loc}^1((\Omega \cup S) \setminus \partial A_{m_h}),\)  \(\widetilde{u}\in GSBD^2({\rm Int}{(\Omega \cup S\cup \Sigma )})\cap H_\mathrm {loc}^1((\Omega \cup S) \setminus \partial G),\)  \(J_{\widetilde{u}}\subset (\Omega \cap \partial G)\cup (\Sigma \cap J_u)\)  and
\(\mathcal {H}^1(J_{\widetilde{u}_h}) + \int _\Omega |e(\widetilde{u}_h)|^2dx\le \mathcal {F}(A_{m_h},u_{m_h}) + \mathcal {H}^1(\Sigma )\le M:= \mathcal {F}(A_1,u_1) + \mathcal {H}^1(\Sigma )<\infty \) 
for every \(h\ge 1.\)  Moreover, by Proposition REF , the definitions of \(u,\)  \(\widetilde{u}_h\)  and \(\widetilde{u},\)
\(\widetilde{u}_h \rightarrow \widetilde{u} \quad \text{a.e. in $[S\cup (\Omega \setminus G)]\cup \bigcup \limits _{i\in N} E_i$}\) 
and
\(|\widetilde{u}_h| \rightarrow +\infty \quad \text{a.e. in $\bigcup \limits _{j\in \mathbb {N}\setminus N} E_j,$}\) 
where \(\lbrace E_i\rbrace \)  and \(N\)  are provided by Proposition REF .
We observe that a priori \(\partial A,\)  and hence \(\partial G\) , does not need to be \(\mathcal {H}^1\) -rectifiable. Therefore, by [1]} \(J_{\widetilde{u}}\subset (\Omega \cap \partial ^r G)\cup (\Sigma \cap J_u),\)  where we recall that \(\partial ^rG\)  is \(\mathcal {H}^1\) -rectifiable part of \(\partial G.\)
To prove (REF ) we use similar arguments as in [2]}. Let
\(g \in L^\infty (\Sigma \times \lbrace 0,1\rbrace )\)  be such that
\(g(x,s): = \varphi (x,\nu _\Sigma (x)) + s\beta (x)\) 
for which we know by (REF ) that \(g\ge 0\)  and
\(|g(x,1) - g(x,0)| \le \varphi (x,\nu _\Sigma (x))\qquad \text{for a.e. $x\in \Sigma .$ }\) 
Let \(\mu _h\)  be the sequence of positive Radon measures defined at Borel sets \(B\subset \mathbb {R}^2\)  as
\(\begin{aligned}\mu _h(B) &: = \int _{B\cap \Omega \cap \partial ^*A_{m_h}} \varphi (x,\nu _{A_{m_h}})d\mathcal {H}^1 + 2\int _{B\cap \Omega \cap (A_{m_h}^{(1)}\cup A_{m_h}^{(0)})\cap \partial A_{m_h}}\varphi (x,\nu _{A_{m_h}})d\mathcal {H}^1\nonumber \\& + \int _{B\cap \Sigma \cap A_{m_h}^{(0)}\cap \partial A_{m_h} } \big [\varphi (x,\nu _\Sigma )+ g(x,1)\big ]d\mathcal {H}^1(x) + \int _{B\cap \Sigma \setminus \partial A_{m_h}} g(x,0)d\mathcal {H}^1 \nonumber \\& + \int _{B\cap \Sigma \cap \partial ^*A_{m_h} \setminus J_{u_{m_h}} } g(x,1) d\mathcal {H}^1+ \int _{B\cap \Sigma \cap J_{u_{m_h}}} \big [g(x,0) + \varphi (x,\nu _\Sigma )\big ]\,d\mathcal {H}^1\end{aligned}\) 
and
let \(\mu \)  be the positive measure defined at Borel sets \(B\subset \mathbb {R}^2\)  as
\(\begin{aligned}& \mu (B) : = \int _{B\cap \Omega \cap \partial ^*G} \varphi (x,\nu _A)d\mathcal {H}^1 +2\int _{B\cap \Omega \cap G^{(1)} \cap \partial G\cap J_{\widetilde{u}}} \varphi (x,\nu _G)d\mathcal {H}^1\\+& \int _{B\cap \Sigma \setminus \partial G} g(x,0)d\mathcal {H}^1 + \int _{B\cap \Sigma \cap \partial ^*G \setminus J_u} g(x,1) d\mathcal {H}^1+ \int _{B\cap \Sigma \cap J_{\widetilde{u}}} \big [g(x,0) + \varphi (x,\nu _\Sigma )\big ]\,d\mathcal {H}^1.\end{aligned}\) 
Since \(S_{\widetilde{u}}^A:=G^{(1)}\cap \partial G\cap J_{\widetilde{u}}\)  and \(\Sigma \cap J_{\widetilde{u}} = \Sigma \cap J_u,\) 
we have
\(\mu _h(\mathbb {R}^2) = \mathcal {S}(A_{m_h} ,u_{m_h}) + \int _{\Sigma } \varphi (x,\nu _\Sigma )d\mathcal {H}^1\) 
and
\(\mu (\mathbb {R}^2) = \widetilde{\mathcal {S}}(G,u) + \int _{\Sigma } \varphi (x,\nu _\Sigma )d\mathcal {H}^1.\) 
Hence, to establish (REF ) it suffices to prove
\(\liminf \limits _{h\rightarrow \infty } \mu _h(\mathbb {R}^2) \ge \mu (\mathbb {R}^2).\) 
Since \(\sup _h \mu _h(\mathbb {R}^2)<+\infty ,\)  by compactness, there exists a positive Radon measure \(\mu _0\)  in \(\mathbb {R}^2\)  such that (up to a subsequence) \(\mu _h\rightharpoonup ^* \mu _0\)  as \(h\rightarrow \infty .\) 
We show
\(\mu _0\ge \mu \) 
and we observe that (REF ) immediately follows from (REF ).
To establish (REF ) it suffices to prove
\(& \frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Omega \cap \partial ^*G)}(x) \ge \varphi (x,\nu _G(x))\qquad \text{for a.e. $x\in \Omega \cap \partial ^*G,$}\\& \frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Sigma \cap \partial ^* G)}(x) \ge g(x,1) \qquad \text{for a.e. $x\in \Sigma \cap \partial ^* G,$}\\& \frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Sigma \setminus \partial G)}(x) \ge \varphi (x,\nu _\Sigma (x)) \qquad \text{for a.e. $x\in \Sigma \setminus \partial G,$} \\& \frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits S_{\widetilde{u}}^A }(x) \ge 2\varphi (x,\nu _G(x))\qquad \text{for a.e. $x\in S_{\widetilde{u}}^A,$ }\\& \frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Sigma \cap J_{\widetilde{u}})}(x) \ge 2\varphi (x,\nu _\Sigma (x)) \qquad \text{for a.e. $x\in \Sigma \cap J_{\widetilde{u}}$}\) 


since \(g(x,0)=\varphi (x,\nu _\Sigma ).\) 
We use similar arguments of the proof of [2]}.
The proof of the estimates (REF )-() follows from similar arguments used in [2]} with special care needed for (). In fact for () we cannot employ the strategy used for [2]} which was hinged on the uniform bound on the number of boundary components, which here we do not have. We instead adapt the arguments employed in [2]} by using Lemma REF .
Next we detail the proofs of (REF )-().
Proof of (REF ). Note that \(A =G\)  up to a negligible set.
By Corollary REF  \(A_{m_h}\rightarrow A\)  in \(L^1(\mathbb {R}^2),\)  thus, the proof of (REF ) can be done following the arguments of [2]} using Reshetnyak lower semicontinuity Theorem [8]}.
Proof of ().
Since \(A_{m_h} \rightarrow G\)  in \(L^1(\mathbb {R}^2),\)  we have \(D\chi _{A_{m_h}}\rightharpoonup ^* D\chi _G.\)  Thus, the proof of () directly follows from [9]} (see also the proof of [2]}).
Proof of ().
Let \(x_0\in \Sigma \setminus \partial G\)  and let \(r_0:=\mathrm {dist}(x_0,\partial G) >0.\)  Since \(\mathbb {R}^2\setminus \mathrm {Int}(A_{m_h})\overset{K}{\rightarrow } \mathbb {R}^2\setminus \mathrm {Int}(A)=\mathbb {R}^2\setminus G,\)  there exists \(r_1\in (0,r_0)\)  such that \(B_r(x_0)\cap \mathrm {Int}(A_{m_h})\)  for any \(r\in (0,r_1).\)  Therefore, for any \(r\in (0,r_1)\)
\(\begin{aligned}\mu _h(\overline{B_r(x_0)}) = & \int _{\overline{B_r(x_0)}\cap \Sigma \cap A_{m_h}^{(0)}\cap \partial A_{m_h} } \big [\varphi (x,\nu _\Sigma )+ g(x,1)\big ]d\mathcal {H}^1(x) + \int _{\overline{B_r(x_0)}\cap \Sigma \setminus \partial A_{m_h}} g(x,0)d\mathcal {H}^1\\\ge & \int _{\overline{B_r(x_0)}\cap \Sigma \cap A_{m_h}^{(0)}\cap \partial A_{m_h} } g(x,0)d\mathcal {H}^1(x) + \int _{\overline{B_r(x_0)}\cap \Sigma \setminus \partial A_{m_h}} g(x,0)d\mathcal {H}^1 \\= & \int _{\overline{B_r(x_0)}\cap \Sigma } g(x,0)d\mathcal {H}^1,\end{aligned}\) 
where in the inequality we used (REF ).
Thus, taking \(limsup\)  as \(h\rightarrow +\infty \)  and using \(\mu _h \rightharpoonup ^*\mu _0\)  we get
\(\mu _0(\overline{B_r(x_0)}) \ge \int _{B_r(x_0)\cap \Sigma } g(x,0)d\mathcal {H}^1.\) 
Now () follows from the Besicovitch Differentiation Theorem.
Proofs of () and (). We establish
\(\frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits K } \ge 2\phi (x,\nu _K) \quad \text{for $\mathcal {H}^1$-a.e. $x\in K,$}\) 
where
\(K=S_{\widetilde{u}}^A \cup (\Sigma \cap J_{\widetilde{u}}).\) 
Let \(x\in K\)  be such that \(\theta (K,x)=1.\)  Then either \(x\in S_{\widetilde{u}} \subset G^{(1)}\cap \partial ^r G\)  or \(x\in \Sigma \cap J_{\widetilde{u}}.\)  By setting \(E_0:=S\)  and recallling that \(\mathrm {Int}(A)=\cup _{i\in \mathbb {N}} E_i,\)  in view of Proposition REF 
we have the one of the following:
(b1)
there exists \(i_0\in N\)  such that \(x\in E_{i_0}^{(1)}\cap \partial ^r E_{i_0},\) 
\(\theta (E_{i_0}^{(1)}\cap \partial ^r E_{i_0},x)=1\) 
and \(u_{m_h} + a_{m_h} \rightarrow u\)  a.e. in \(E_{i_0};\)
(b2)
there exist \(i_0\in N\cup \lbrace 0\rbrace \)  and \(j_0\in \mathbb {N}\setminus N\) 
such that \(x\in \partial ^*E_{i_0}\cap \partial ^* E_{j_0}\)  and \(u_{m_h} + a_{m_h}\rightarrow u\)  a.e. in \(E_{i_0}\)  and \(|u_{m_h} + a_{m_h}|\rightarrow \infty \)  a.e. in \(E_{j_0};\)
(b3)
there exist \(i_1,i_2\in N\cup \lbrace 0\rbrace \)  with \(i_1\ne i_2\)  such that \(x\in \partial ^*E_{i_1}\cap \partial ^* E_{i_2}\)  and \(u_{m_h} + a_{m_h}\rightarrow u\)  a.e. in \(E_{i_1}\cup E_{i_2}.\)

Let \(L\)  denote the set among \(E_{i_0}^{(1)}\cap \partial ^r E_{i_0},\)  \(\partial ^*E_{i_0}\cap \partial ^* E_{j_0}\)  and \(\partial ^*E_{i_1}\cap \partial ^* E_{i_2}\)  containing \(x.\)  Without loss of generality we assume that \(x\in Y\subset L,\) 
where \(Y\)  is defined as the set of points \(y\in L\subset \partial A\)  satisfying:
(c1)
\(\theta (\partial G,y) = \theta (\partial A,y) =\theta (L,y)= 1\)  and \(\nu _G(y)=\nu _A(y)=\nu _L(y)\)  exists. If \(y\in \Sigma ,\)  then additionally, \(\theta (\Sigma ,x) =1\)  and \(\nu _\Sigma \)  also exists;
(c2)
as \(\rho \rightarrow 0\)  the sets \(\overline{ Q_{R,\nu _L}(y)}\cap \sigma _{\rho ,x}(\partial A), \)  \(\overline{ Q_{R,\nu _L}(x)}\cap \sigma _{\rho ,y}(\partial G) \)  and \(\overline{ Q_{R,\nu _L}(y)}\cap \sigma _{\rho ,y}(L) \)  converge \(\overline{Q_{R,\nu _L}(y)} \cap T_y\)  in the Kuratowski sense, where \(R>0\)  and \(T_y\)  is the generalized tangent line to \(\partial A\)  at \(y;\)
(c3)
one-sided traces \(\widetilde{u}^+(y)\)  and \(\widetilde{u}^-(y)\)  of \(\widetilde{u}\)  w.r.t. \(L\)  exist and are not equal;
(c4)
\(\frac{d\mu _0}{ d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits K}\,(y)\)  exists and is finite.

In fact, \(\mathcal {H}^1(L\setminus Y)=0\)  since for (c1) we notice that \(Y\subset L \subset \partial ^rA\)  and \(\partial ^rA\)  is \(\mathcal {H}^1\) -rectifiable, for (c2) we use Proposition REF  by observing that the points of \(\Sigma \)  and \(\Omega \cap \partial A\)  satisfy uniform density estimates in view of the Lipchitzianity of \(\Sigma \)  and Proposition REF , respectively, for (c3) we use
[1]} and the existence of traces of \(GBD\) -functions along \(C^1\) -manifolds [1]} and the fact that being a jump set of \(\widetilde{u},\)  the set \(K\)  (and also \(L\) ) can be covered by at most countably many one-dimensional \(C^1\) -graphs (up to a \(\mathcal {H}^1\) -negligible set), and finally for (c4) we use Besicovitch Differentiation Theorem.
Without loss of generality, we assume \(x=0,\)  \(\nu _K(x)={\bf e_2},\)  \(T_x\)  is the \(x_1\) -axis and \({\bf e_2}\)  is the outer normal of \(E_{i_0}\) .
Let \(4r_0:=\mathrm {dist}(0,\partial \Omega )\)  if \(0\in \Omega \)  and \(4r_0:=\mathrm {dist}(0,\partial \Sigma )\)  if \(0\in \Sigma ;\)  since \(\Sigma \)  is Lipschitz, it consists of at most countably many open connected components in \(\partial \Omega ,\)  and hence, \(r_0>0\) . By weak convergence,
\(\lim \limits _{h\rightarrow \infty } \mu _h(\overline{ Q_r}) = \mu _0( Q_r)\) 
for a.e. \(r\in (0,r_0).\)  By assumption (b3), [1]} and [1]} separately applied to \( Q_1^+:=Q_1\cap \lbrace x_2>0\rbrace \)  and \( Q_1\setminus Q_1^+\)  we have
\(\lim \limits _{r\rightarrow 0} \int _{ Q_1} |\tau (\widetilde{u}(rx)) - \tau (u_0(x))| dx =0,\) 
where
\(u_0 := \widetilde{u}^+(0)\chi _{ Q_1^+} + \widetilde{u}^-(0)\chi _{ Q_1\setminus Q_1^+}\) 
and
\(\tau (z) = (\arctan z_1,\arctan z_2),\quad z=(z_1,z_2)\in \mathbb {R}^2.\) 
For every \(r\in (0,r_0)\)  let
\(U_r^\infty :=\lbrace x\in Q_1:\,\, \liminf \limits _{h\rightarrow \infty } |\widetilde{u}_h(rx)|=+\infty \rbrace .\) 
Unlike the proof of [2]}, (REF ) implies that \(U_r^\infty \)  can have positive measure.
By (REF ) and the Dominated Convergence Theorem
\(\lim \limits _{h\rightarrow \infty } \int _{ Q_1\setminus U_r^\infty } |\tau (\widetilde{u}_h(rx)) - \tau (\widetilde{u}(rx))| dx = 0.\) 
By (c2) applied with \(R=8\) , Proposition REF  and (c1)-(c3)
\(& Q_8\cap \sigma _r(\partial A) \overset{K}{\rightarrow } I_8\qquad \text{and}\qquad \mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits ( Q_8\cap \sigma _r(\partial A)) \overset{*}{\rightharpoonup } \mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits I_8, \\& Q_8\cap \sigma _r(L) \overset{K}{\rightarrow } I_8\quad \text{and}\quad \mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits ( Q_8\cap \sigma _r(L) )) \overset{*}{\rightharpoonup } \mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits I_8 \) 

as \(r\rightarrow 0.\)  Hence, by [2]}
\(& \mathrm {sdist}(\cdot , \sigma _r(\partial A)) \rightarrow - \mathrm {dist}(\cdot ,T_0), \\& \mathrm {sdist}(\cdot , \sigma _r(\partial E_{i_0})) \rightarrow -\mathrm {dist}(\cdot ,T_0),\\&\mathrm {sdist}(\cdot , \sigma _r(\partial [E_{i_0}\cup E_{j_0}])) \rightarrow -\mathrm {dist}(\cdot ,T_0), \\&\mathrm {sdist}(\cdot , \sigma _r(\partial [E_{i_1}\cup E_{i_2}])) \rightarrow -\mathrm {dist}(\cdot ,T_0) \) 


locally uniformly in \(\overline{ Q_4}\)  as \(r\rightarrow 0.\) 
Let
\(U_0^\infty ={\left\lbrace \begin{array}{ll}\emptyset & \text{in cases (c1) and (c3)},\\Q_1^+ & \text{in case (c2)}.\end{array}\right.}\) 
By the definitions of \(E_{i_0},\)  \(E_{j_0},\)  \(E_{i_1}\)  and \(E_{i_2}\)  and ()-()
\(\lim \limits _{r\rightarrow 0} | U_r^\infty \Delta U_0^\infty | =0.\) 
Step 1: We choose sequences \(h_k\nearrow \infty \)  and \(r_k\searrow 0\)  as follows. By (REF ), (REF ), (REF ) and (REF ) for any \(k\in \mathbb {N}\)  there exists \(r_k\in (0,\frac{1}{k})\)  such that (REF ) holds with \(r=r_k\)  and
\(&\Vert \mathrm {sdist}(\cdot , \sigma _{r_k} (\partial A)) + \mathrm {dist}(\cdot , T_0) \Vert _{L^\infty ( Q_4)} <\frac{1}{k^2},\\&\int _{ Q_1} |\tau (\widetilde{u}(r_k x)) - \tau (u_0(x))| dx <\frac{1}{k^2}, \\&| U_{r_k}^\infty \Delta U_0^\infty | < \frac{1}{k^2} .\) 

Given \(k\ge 1\)  and \(r_k\) , since \(A_{m_h}\)  \(sdist\) -converges to \(A\)  and the function \(\tau \)  is bounded, by (REF ), () and (REF ) we can choose \(h_k\)  such that
\(&\frac{1}{h_kr_k} <\frac{1}{k},\\&\Vert \mathrm {sdist}(\cdot , \sigma _{r_k} (\partial A_{m_{h_k}} )) - \mathrm {sdist}(\cdot , \sigma _{r_k} ( \partial A))\Vert _{L^\infty ( Q_4)} <\frac{1}{k},\\&\int _{ Q_1\setminus U_0^\infty } |\tau (\widetilde{u}_{h_k}(r_kx)) - \tau (\widetilde{u}(r_kx))| dx < \frac{1}{k},\\& \mu _{h_k}(\overline{ Q_{r_k}}) \le \mu _0( Q_{r_k}) +r_k^2. \) 


Notice that by (REF ), \(h_k\rightarrow \infty \)  as \(k\rightarrow \infty .\)
Let
\(D_k:= \sigma _{r_k} (A_{m_{h_k}}\cup S)\) 
and
\(w_k(x):=\widetilde{u}_{h_k}(r_kx),\quad x\in Q_1.\) 
Then the number of connected components of \(\partial D_k\)  lying strictly inside \(Q_4\)  does not exceed \(m_{h_k},\)  and \(w_k\in GSBD^2(Q_1)\)  with \(J_{w_k}\subset Q_1\cap \partial D_k.\) 
By ()
and (REF ),
\(\mathrm {sdist}(\cdot ,\partial D_k) \rightarrow -\mathrm {dist}(\cdot , T_0) \quad \text{uniformly in $Q_4$ as $k\rightarrow \infty .$}\) 
Moreover, by () and () \(w_k\rightarrow u_0\)  a.e. in \(Q_1\setminus U_1^\infty \)  and \(|w_k|\rightarrow +\infty \)  a.e. in \(U_1^\infty .\) 
By the finiteness of
\(\frac{d\mu _0}{\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits L}(0) = \lim \limits _{k\rightarrow \infty } \frac{\mu _0( Q_{r_k})}{2r_k}\) 
and ()
\(\mathcal {H}^1( Q_1\cap \partial D_k) =\frac{\mathcal {H}^1( Q_{r_k}\cap \partial A_{m_{h_k}} )}{r_k} \le \frac{\mu _{h_k}(\overline{ Q_{r_k}})}{c_1 r_k} \le C:=\frac{2}{c_1}\frac{d\mu _0}{\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits L}(0) + 1\) 
for all large \(k.\) 
Moreover, by changing variables as \(x=r_ky\)  and using (REF ) we get
\(\int _{Q_1} |e(w_k)|^2dx = \int _{Q_{r_k}} |e(\widetilde{u}_k)|^2dy \le M\) 
for all \(k;\)  note that the first equality holds only in dimension two.
Fix \(\delta \in (0,1).\) 
Since \(\varphi \)  is uniformly continuous, there exists \(k_\delta ^0>0\)  such that
\(|\varphi (x,\nu ) - \varphi (0,\nu )| < \delta ,\qquad x\in \overline{ Q_{r_k}},\,\,\nu \in \mathbb {S}^1.\) 
Therefore, by the definitions of \(D_k\)  and \(\mu _h,\)  the nonnegativity of \(g\)  as well as (REF )
\(\frac{\mu _{h_k}(\overline{ Q_{r_k}})}{r_k} \ge &\int _{ Q_1\cap \partial ^* D_k} \phi (\nu _{D_k})d\mathcal {H}^1+ 2\int _{ Q_1\cap D_k^{(1)} \cap \partial D_k} \phi (\nu _{D_k})d\mathcal {H}^1 - 2Cc_2\delta ,\) 

where
\(\phi (\nu ) = \varphi (0,\nu ).\) 
By Lemma REF  applied with sequences \(\lbrace D_k\rbrace \)  and \(\lbrace m_{h_k}\rbrace \)  we find \(k_\delta ^2>k_\delta ^1\)  such that
\(\int _{ Q_1\cap \partial ^* D_k} \phi (\nu _{D_k})d\mathcal {H}^1+ 2\int _{ Q_1\cap D_k^{(1)} \cap \partial D_k} \phi (\nu _{D_k})d\mathcal {H}^1 \ge 2\int _{I_1} \phi ({\bf e_2})d\mathcal {H}^1 - \delta .\) 
Thus, by (REF ) and () we get
\(\frac{\mu _0(Q_{r_k})}{2r_k} + \frac{r_k}{2} \ge \int _{I_1} \phi ({\bf e_2})d\mathcal {H}^1 - \frac{2Cc_1 + 1}{2}\,\delta \) 
for all \(k>k_\delta ^2.\)  Now letting first \(k\rightarrow +\infty \)  and then \(\delta \rightarrow 0\)  we get (REF ).
